Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$

The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations.

💡 Research Summary

The paper establishes a relative version of the Kazhdan–Lusztig (KL) isomorphism for the symmetric pair ((GL_{2n},Sp_{2n})). The classical KL isomorphism identifies the affine Hecke algebra (\mathcal H(G,I)) of a (p)-adic group (G) with the equivariant (K)-theory of the Steinberg variety of its complex Langlands dual (G^\vee). This identification underlies the proof of the Deligne–Langlands conjecture, which classifies smooth irreducible representations of (G) possessing an Iwahori‑fixed vector via triples ((t,n,\chi)).

The author adapts this framework to the symmetric space (X=GL_{2n}/Sp_{2n}). First, the Hamiltonian space (M=T^*X) is considered, and following the relative Langlands duality of Ben‑Gurion–Zhang–Venkatesh, a dual Hamiltonian space (M^\vee) is constructed. Concretely, one introduces the subgroup \